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Among the different generalizations of the CLT available on the web, I found these

  • CLT for the sum of non-identical (and independent) random variables
  • CLT for the sum of identical (and independent) multivariate random variables.

However, I can't find any for the sum of non-identical (and independent) multivariate variables.

Is it because it is straightforward ? Would the final covariance matrix be simply the sum of the individual ones ?

Thanks

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  • $\begingroup$ Why is this off-topic? Is it because it should be on math.stackexchange instead? $\endgroup$
    – bob
    May 18, 2022 at 20:49

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I always prefer to have error bounds for the CLT, so my favorite reference for your question is the paper "A Lyapunov type bound in $\mathbb{R}^d$" by Vidmantas Bentkus (Theory of Probability & Its Applications 49(2), 311--323, 2005).

From the abstract: Let $X_1, \dots, X_n$ be independent, mean-zero, $\mathbb{R}^d$-valued random variables. Let $S = X_1 + \cdots + X_n$ and let $C^2$ be the covariance matrix of $S$, assumed invertible. Let $Z$ be a $d$-dimensional Gaussian with mean zero and covariance $C^2$. Then for any convex subset $A \subseteq \mathbb{R}^d$,

$$|\Pr[S \in A] - \Pr[Z \in A]| \leq O(d^{1/4}) \cdot \beta,$$ where $\beta = \sum_{i} \mathbf{E}[|C^{-1}X_i|^3]$. This is a $d$-dimensional generalization of the Berry--Esseen Theorem.

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    $\begingroup$ I kind of agree with the (now-deleted) comment made by OP that it's not clear why this question is On Hold. It's not so easy to find clear statements about the multidimensional, non-iid, CLT in the literature. For example, the (first?) paper on the multidim CLT by Sazonov only treats the iid case, and the standard textbook by Bhattacharya and Rao is a bit of a nightmare to read (in my opinion). $\endgroup$
    – Ryan O'Donnell
    Jul 18, 2013 at 20:51
  • $\begingroup$ Thank you! I have had a hard time figuring out the bounds in Gotze and others. Do you know if there is also a multiplicative bound? Something like $-\log\Pr[S\in A] = L(A) x^2 n + O(x^3) + o(n)$ where $L$ is some function and $x$ is some distance parameter from $A$ to the origin. It seems to me a bound can be derived from Large Deviation Theory, but I can't find any good references. $\endgroup$
    – Thomas Dybdahl Ahle
    Jun 22, 2017 at 1:27

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